Seven surface GHG monitoring stations were selected as a test bed for evaluating the inverse modelling approach. These seven stations (Table 1) in Alberta, Saskatchewan and Ontario represent regions which include the major Canadian anthropogenic GHG emissions. In 2012, CO2 contributed 79 % of Canada's total GHG emissions that were estimated to be 699 megatonnes (Mt) of CO2 equivalent (Environment Canada, 2012). The majority of these emissions resulted from the combustion of fossil fuels and the remaining portions were contributed from industrial processes and waste incinerations. These seven GHG stations are located in the three Canadian provinces that contributed close to 70 % of the national total GHG emissions annually according to the GHG national inventory of Canada (Environment Canada, 2012).

In this study, the inversion was done separately for the western region of Alberta and Saskatchewan provinces and the eastern region of Ontario. As shown in Fig. 2a–g, seven region definitions were used in this study to define sub-regions to be optimized; (2a) two sub-regions for the provinces of Alberta/Saskatchewan (AB/SK), one sub-region for the province of Ontario (ON), (2b) four sub-regions for AB/SK, two sub-regions for ON, (2c) seven sub-regions for AB/SK, four sub-regions for ON, (2d) eleven sub-regions for AB/SK, six sub-regions for ON, (2e) nineteen sub-regions for AB/SK, twelve sub-regions for ON, (2f) twenty-seven sub-regions for AB/SK, twenty-four sub-regions for ON, and (2g) thirty-seven sub-regions (census divisions) for AB/SK and forty-nine sub-regions (census divisions) for ON to investigate whether there are problems and/or benefits (magnitude of biases and uncertainties) of estimating a large number of parameters.

Two sets of fossil fuel CO2 fluxes were used as prior and target (known “truth”) fluxes and summarized in Table 2, which includes the monthly and annual provincial totals from CT2010 and CT2011 fossil fuel CO2. The fluxes were uniformly re-distributed to 0.2∘×0.2∘ from the original resolution of 1∘×1∘ to be folded into the emission sensitivity fields from FLEXPART (next section). For visualization, the gridded fluxes were aggregated into sub-regions to be optimized as shown in Fig. 2. Year 2009 country and global totals (by fuel type) were extrapolated from the 2007 Carbon Dioxide Information Analysis Center (CDIAC, Boden et al., 2013) used for the CT2010 fossil fuel fluxes (CarbonTracker, 2010). Open-source Data Inventory for Anthropogenic CO2 (ODIAC, Oda and Maksyutov, 2011) emissions are spatially distributed using many available “proxy data” that explain spatial extent of emissions according to emission types (emissions over land, gas flaring, aviation and marine bunker). CarbonTracker combined the ODIAC emissions with CDIAC emissions to generate CT2011 fossil fuel fluxes (Andres et al., 2011; CarbonTracker, 2011). Note that CT2011 is half CDIAC plus half ODIAC, and ODIAC has different spatial distribution than CDIAC.

The European Centre for Medium-range Weather Forecasts (ECMWF) operational wind fields at T799 spectral resolution were used to drive the dispersion model of FLEXPART. The ECMWF modelled data were retrieved with a temporal resolution of 3 h (analyses at 00:00, 06:00, 12:00, and 18:00 UTC; forecasts at 03:00, 09:00, 15:00, and 21:00 UTC), interpolated to horizontal resolution of 0.2∘×0.2∘ on the Gaussian grid over Canada and the US (180∘ W to 0∘ E and 20 to 90∘ N) nested in a global grid with resolution of 1∘×1∘. Both grids had 91 vertical levels. The FLEXPART Lagrangian particle dispersion model (Stohl et al., 2005) was used to simulate the 5 day transport history (retroplume) of the fossil fuel CO2 mole fractions at each station location. The model calculated the trajectories of 5000 particles daily at 21:00 UTC (14:00 to 16:00 LST depending on time zones) from the intake height at each station location being considered.

Retroplume spatial distributions were output as 30 min averages on a 0.2∘×0.2∘ degree grid. The retroplumes were then summed up for the entire 5 days for each time point of particle release. FLEXPART outputs the retroplumes in units of skg-1m3, which is the residence time of the plume per grid cell divided by the air density. The footprint layer of the retroplume for FLEXPART is fixed at the standard 100 m layer adjacent to the Earth's surface (Stohl et al., 2005). The modelled fossil fuel CO2 mole fractions were constructed by multiplying the footprint layer with the monthly prior fossil fuel CO2 fluxes at 0.2∘×0.2∘ in kgs-1 and summed up over all grid cells (plus the baseline or the contribution from prior to the 5 day simulation period, described below) to yield the time series of modelled fossil fuel CO2 mole fractions at the measurement station (Stohl et al., 2003, 2009; Cooper et al., 2010).

The station-specific baseline in this context is considered the influence from outside the continent far from the inversion domain. The mole fractions of the fossil fuel CO2 were sampled from the CT2011 predicted global fossil fuel CO2 field at the positions (latitude, longitude and altitude) of 5000 particles at the end of the 5th day backward simulation for each station released at 21:00 UTC daily to obtain 5000 mole fraction values. These 5000 mole fractions were averaged to represent the mean baseline for each release time point. The station-specific baseline time series was subsequently subtracted from the synthetic observations that were sampled from CT2011 for each station. This allowed us to infer fluxes over the region of interest. Errors in the baseline estimation in this study were treated as a part of the transport error when CT2011 mole fractions were used as the “target”.

In addition to the more commonly known CFM approach, we include a simulation-based method for parameter estimations, MCMC. The performances of the two inversion methods in terms of percentage differences between the posterior estimates and the target fossil fuel CO2 fluxes are assessed. Note that matrices and vectors are in bold and italic throughout this paper, whereas scalar quantities are in italic font. Inversion was done separately for the western and eastern domains, and for each month of 2009.

The prior gridded fluxes of fossil fuel CO2, {xg,p,t} were re-distributed from the original 1∘×1∘ uniformly to the same spatial resolution of 0.2∘×0.2∘ as the emission source sensitivities (or footprints), {Mg,p,t,s} where the subscripts are, g for a given grid cell in space, sub-region p, station s and time t. xg,p,t is the gridded emission field over sub-region p at time t. The footprints vary in space, time and stations. The modelled mole fractions in our experiments were limited to 21:00 UTC daily (14:00 to 16:00 LST depending on time zones) in January through December for 2009 to avoid temporal correlation and night time processes. Two regions of interest are the two neighboring provinces of Alberta and Saskatchewan (western region), and separately, the province of Ontario (eastern region) in Canada. Any remaining contributions from outside of the inversion region but within the 5 day integration period were subtracted from the synthetic observations for each station in addition to the station-specific baseline time series.

Using a simple linear regression model (likelihood function), linear scaling factors λp for xg,p,t are estimated to fit the synthetic observations yt,s. The potential advantage of this linear regression model is that fewer assumptions are needed compared to the typical CFM method (Sect. 2.7). This simple regression method is compared to the CFM method to evaluate these two approaches. The regression model is shown below: yt,s=∑p∈Rλp∑g∈GMg,p,t,sxg,p,t+ϵt,s for station s, at time t, scaling factors λp for sub-region p to be estimated, Mg,p,t,s is the station-specific emission sensitivity (footprint) to be summed up over the sub-region p for each FLEXPART footprint grid cell g with G being the total number of grid cells of sub-region p. For a given time t and station s, summing contributions from all sub-regions to the total number of R sub-regions gives the total modelled mole fraction. Let Kp,t,s=∑g∈GMg,p,t,sxg,p,t be the contribution from sub-region p, for station s at time t. We obtain: yt,s=∑p∈RλpKp,t,s+ϵt,s

In the MCMC simulation method, same prior error variances (σprior)2 and (σe)2 for λ (R×1 vector) and ϵ (N×1 vector, where N equal to the total number of synthetic observations which equal to the number of time points times the number of stations) are used as in the CFM method, but the posterior estimates and the uncertainties for λ are calculated by drawing samples from the joint distributions of the log likelihood and the assumed distributions of prior parameters λprior (briefly described below) instead of solving for the parameters analytically. See the detailed description in the Appendix.

To implement the regression model Eq. (1), we consider the following Bayesian inversion settings for the western region and the eastern region. Assume λp follows normal distribution with a mean of 1 and a variance of 1 for (σprior)2, which corresponds to 100 % error. In the MCMC method, (σe)2 is assumed to follow inverse-gamma distribution, the mean and variance for (σe)2 are prescribed by setting the shape and scale parameters to 2.1 and 1.1 respectively. This gives a mean of 1 and a variance of 10. Sensitivity analysis is performed in the synthetic data experiments (E1, E2 and E4), in which the shape and scale parameters (Appendix) are changed to 2.001 and 1.001 respectively. This gives a mean of 1 and a variance of 1000 for the (σe)2, which correspond to conjugate non-informative priors. Using non-informative priors allows MCMC to sample parameter estimates from a wide parameter space (Appendix).

In our MCMC simulation, a random-walk Metropolis algorithm (Appendix) (Roberts, 1996; Liu, 2001) was used to obtain posterior scaling factor estimates for the sub-regions. The λp is initialized to 1, and each monthly batch inversion has 110 000 iterations (first 10 000 discarded as burn-in samples), thinning rate is set to every 10th (every 10th drawn vector of scaling factor estimates is kept), the number of simulations saved for subsequent inferences is equal to 10 000 for each month. Although the use of mean posterior estimates should be avoided (Tarantola, 2005), it is necessary here to compare the results using MCMC to those using the CFM method. Subsequently, the monthly posterior provincial total flux estimates are calculated using the mean of 10 000 scaling factors simulated by the MCMC procedure multiplied by the prior fluxes as shown in Eq. (3). SAB=∑p=1Rλp,ABχp,ABSSK=∑p=1Rλp,SKχp,SKSON=∑p=1Rλp,ONχp,ON where SAB, SSK and SON are the monthly posterior provincial total fossil fuel CO2 fluxes for Alberta (AB), Saskatchewan (SK), and Ontario (ON) respectively. Note that λp,AB, λp,SK and λp,ON are the mean scaling factors of the sub-regions within the respective province simulated by the MCMC method. χp,AB, χp,SK and χp,ON are the monthly prior fluxes for sub-region p in the respective province. With large number of simulated scaling factors, various statistics on the posterior provincial total fluxes can be calculated such as the percentiles, standard deviations and 95 % confidence intervals.

The optimal posterior estimates of scaling factors are obtained by minimizing the cost function J, J(λ)=(y-Kλ)TDϵ-1(y-Kλ)+(λ-λprior)TDprior-1(λ-λprior) where y(N×1) is the vector of observations (synthetic observations). λ(R×1) is the vector of the posterior scaling factors to be estimated, N= number of time points times number of stations, λprior is the vector of the prior scaling factors which are all initialized to 1 for all sub-regions and K(N×R) is the matrix of contributions from different sub-regions. K is the product of two matrices, M and χ. M is the modelled transport (or footprints in our case) and χ is the spatial distribution of the surface emission fluxes. A linear regularization term has been added which is the second term on the right hand side of Eq. (4), a typical setup for undetermined (under-constrained due to lack of observations) problems such as atmospheric flux inversion.

The estimate for λ is calculated according to the expression below. λ=KTDϵ-1K+Dprior-1-1KTDϵ-1y+Dprior-1λprior where Dprior(R×R)=(σprior)20…00(σprior)2…0⋮⋮⋱⋮00…(σprior)2 and Dϵ(N×N)=(σe)20…00(σe)2…0⋮⋮⋱⋮00…(σe)2

The posterior error variance-covariance, ∑post, for the estimates of λ is calculated according to: ∑post=KTDϵ-1K+Dprior-1-1

The error covariance matrices are typically not known, consequently Dϵ and Dprior are often assumed to be diagonal matrices as presented in Eqs. (6) and (7), where Dϵ is the prior model-observation error diagonal matrix that the diagonal elements are (σe)2 and off-diagonal elements are all zeros. This means that the model-observation mismatch errors are assumed to be uncorrelated. Similarly, Dprior is the prior scaling factor diagonal matrix where the diagonal elements are (σprior)2 and zeros everywhere else. It means that the contributions from the sub-regions are assumed to be uncorrelated. For further  simplification,  same  individual  (σe)2 scalar element is assigned to all measurement stations at all time points. Similarly, same individual (σprior)2 is assigned to all sub-regions being optimized regardless of the variability and magnitude of the source strength from different sub-regions.

Note that the symbols of the individual elements of yt,s, λp, Mg,p,t,s, χg,p,t, Kp,t,s for the MCMC method presented in Eqs. (1) and (2) are consistent with the matrix notations used in Eq. (4) y, λ, M, χ, K for the CFM method. Inversion was done for each month separately, and western and eastern domains separately. Results will be shown for each month of 2009 as well as the annual total.

To have a measure of the ability and limitations of the proposed inversion approaches, five components were examined in this study: (1) the magnitude and spatial distribution of the prior fluxes, (2) modelled transport, (3) number of sub-regions (unknowns to estimate), (4) prior variances of the model-observation error (σe)2 and of the fluxes (σprior)2 and (5) inversion methods to estimate the parameters (scaling factors) for the purpose of assessing the sensitivity introduced by each component.

We conducted a series of inversion experiments presented in Table 3 using different combinations of the five components mentioned previously. The results of the experiments using simulated mole fractions of fossil fuel CO2 for the year 2009 should reveal whether the provincial annual and monthly total CO2 fluxes and the spatial distributions could be retrieved by the inversion approaches with an acceptable degree of statistical confidence.

The experiments progress with increasing deviations from the target fluxes/transport, as shown in Table 3. E1–E31 and E32–E62 correspond to the two estimation methods of MCMC and CFM, respectively.

Table 3a shows the first (I) set of experiments E1–E10 and E32–E41 that represent the idealized conditions in which transport model error, baseline error, and flux error do not exist. The differences in these experiments are the number of sub-regions to be optimized and the assumed model-observation mismatch (σe)2 and flux (σprior)2 errors. The prior and the target modelled mole fractions were simulated using the same fluxes of CT2011 fossil fuel CO2 folded with the footprints from FLEXPART and therefore, transport model and flux errors do not exist.

To contrast, Table 3b shows the second (II) set of experiments E11–E17 and E42–E48, that the CT2010 fossil fuel CO2 fluxes were used instead to simulate the prior mole fractions for each station. In this set of experiments, small flux error is introduced (only within the provincial inversion domains, Table 2), but modelled transport remains perfect.

Table 3c shows the third set (III), E18–E24 and E49–E55 that were used to assess the impact of transport model error alone on the estimated fluxes. This is achieved by simulating the prior mole fractions in FLEXPART and sampling the target mole fractions (synthetic observations) modelled by CT2011 with the baseline mole fractions subtracted (see Sect. ). Both FLEXPART and CarbonTracker used the same set of CT2011 monthly fossil fuel CO2 fluxes.

Lastly, Table 3d shows the fourth (IV) set, E25–E31 and E56–E62 that were used to assess the combined impacts of transport model and flux errors on the estimated fluxes. This is achieved by simulating the prior mole fractions in FLEXPART using the CT2010 monthly fossil fuel CO2 fluxes and sampling the target mole fractions (synthetic observations) from CT2011 which uses the CT2011 monthly fossil fuel CO2 fluxes. This set of experiments represents more realistic scenarios in which transport and flux errors exist and the experiments can be considered similar to inversions using real observations, but possibly with smaller errors. Note that the transport model error includes errors in the simulated synoptic variability by the FLEXPART model and in the baseline mole fractions sampled from the CT2011 using the 5th day end-points of the FLEXPART particle locations.

The results from the first (I) set of experiments E1–E10 (MCMC method) and E32–E41 (CFM method) reveal any inherent errors introduced by the MCMC and the CFM methods respectively. This set of experiments represents the idealized conditions in which transport model and flux errors do not exist. Figure 4 (and Tables S1–S4) shows that there is essentially zero percentage difference (red squares) from the target for the western and eastern regions regardless of any number of sub-regions and the magnitude of the assumed error variances for (σe)2 and (σprior)2 when the MCMC simulation method is used. Although not shown, the percentage error tends to be larger for the sub-regions with small fluxes. Thus the inversion seems to sensibly provide better constraint for sub-regions with strong signals at the measurement stations.

Experiments E32–E41 show that the CFM inversion procedure (in its “standard” version) has strong positive posterior errors. As shown in Fig. 4, the posterior error increases (red squares) as the number of sub-regions increases for both western and eastern regions. These posterior errors are the results of the optimisation scheme, as the prior fluxes and modelled transport are “perfect”. The optimisation scheme does contain an important approximation, namely the error covariance matrices Dϵ and Dprior being diagonal matrices with the off-diagonal elements set to 0 (Eqs. 6 and 7) as the error covariances (cove) and (covprior) are not generally known. The effect of this approximation and the relative roles of the two error matrices are illustrated in the experiments E32–E41. Note that throughout the experiments E32–E41, the observational constraint error matrix Dϵ has elements ranging from 28×S by 28×S to 31×S by 31×S (corresponding to the number of daily (afternoon) synthetic observations for each monthly inversion), where S is the number of stations.

In E32 for Ontario (eastern region), with only 1 sub-region and the prior constraint error matrix Dprior is reduced to 1 by 1 with no off diagonal elements (and no prior covariance error), the inversion procedure works (for all 12months) with negligible error. This indicates that the diagonal matrix approximation for the large matrix Dϵ did not produce significant posterior errors in the inversion results, and the assumption of no error covariance among the daily (synthetic) observational constraints appears to be reasonable.

However, E32 for Alberta/Saskatchewan (western region) which has 2 sub-regions, the inversion posterior error is 13 % annually. Monthly error can range up to 36 %. The prior constraint error matrix Dprior is a 2 by 2 matrix with two off-diagonal elements (with the same value by symmetry). The zero assumption for this off-diagonal element resulted in large error in the inversion, indicative of the significant role the prior flux covariance has on the inversion. The result is similar to the 2 sub-regions used in Ontario (eastern region E33), with 18 % posterior error annually.

The relative importance of model-observation constraint and prior flux constraint was examined further in experiments E33–E36 with 4 sub-regions for AB and SK, and 2 sub-regions for ON. E33 applied comparable weighting (100 %, 100 %) to both (σe)2 and (σprior)2 in the Dϵ and Dprior matrices. E34 (30 %, 30 %) assigned comparable but small uncertainties for the inversion. E35 (100 %, 30 %) assigned stronger constrain to the prior flux, while E36 (30 %, 100 %) assigned stronger constrain to the (synthetic) observations (smaller uncertainty at 30 %).

The results are clear on the relative roles of the error matrices on the inversion results. Percentage posterior errors in the inversion results progressively increased with relative weighting on the prior constraint (E36 to E34 to E35), the annual posterior errors for the western region of AB/SK ranged from 6 % (E36) to 20 % (E34) to 51 % (E35) shown in Fig. 4a (Table S2). The results are similar for the eastern region of ON (7 to 18 to 28 %) as shown in Fig. 4b (Table S4), the same pattern can be seen on the monthly basis. The missing prior sub-region covariance constraint has strong impact on the inversion results. Conversely the missing observational covariance constraint has little impact on the inversion results. Experiment E34 (30 % and 30 % assigned to (σe)2 and (σprior)2) again shows that it is the relative weighting of the error matrices that is important, as the results are essentially the same as E33 (100 % and 100 % to (σe)2 and (σprior)2). The 30 % prior uncertainties in E34 are comparable to similar regional inversion studies, e.g. Gerbig et al. (2003), Zhao et al. (2009), etc.

For the CFM inversion, posterior error increases with the number of sub-regions. The rate of increase is dependent on the relative weighting of the prior uncertainties (σe)2 and (σprior)2. Experiments E36, E37, E38, E39, E40 and E41 used 30 % to (σe)2 and 100 % to (σprior)2, with the number of sub-regions for the AB/SK increasing from 4, 7, 11, 19, 27, 37 respectively, and ON from 2, 4, 6, 12, 24, 49 respectively. The reason for this choice of prior variance values, 30 % for (σe)2 and 100 % for (σprior)2 will be explained more clearly following the discussion on the model sensitivity to the number of sub-regions used in the optimisation next.

As shown in Fig. 4a and b, the inversion posterior errors generally increase with the number of sub-regions, with the corresponding increase in the sub-region prior constraint error matrix size (and corresponding increase in the number of off-diagonal matrix elements set to zero). The annual posterior errors are all positive ranging from 6 to 22 % and 7 to 42 % for AB/SK and ON respectively, using the CFM method. For CFM, posterior error increases with the number of sub-regions, rate of increase is dependent on the relative weighting of the prior and observational constraint uncertainties.

These results indicate the CFM optimization procedure itself can have substantial errors on account of the incomplete information in the cost function, easily reaching 22–42 % depending on the inversion setup (sub-region definitions, measurement sites distribution, etc.). This uncertainty component of the inversion has generally been ignored in previous studies. As this uncertainty is mainly a function of the prior constraint component of the cost function, it should be included in the prior error variance (σprior)2 for completeness. Therefore, combining this optimization procedure uncertainty with the typical emission inventory uncertainty of ∼30% (e.g. Environment Canada, 2012), it appears reasonable to set (σprior)2 to 100 % (or greater since all these uncertainties are poorly known) as in this study. These prior uncertainty settings 30 % for (σe)2 and 100 % for (σprior)2 are used in all the remaining sensitivity experiments.

In summary, there appears to be an inherent increasing systematic positive bias in the posterior flux estimates introduced by the inversion method of CFM, as the number of sub-regions increases, whereas, posterior errors do not depend on the number of sub-regions using the MCMC method when flux and transport model errors do not exist.

The second (II) set of experiments E11–E17 (MCMC method with increasing number of sub-regions) and E42–E48 (CFM method) represents conditions in which there is no transport model error, but only flux error exists. The prior flux is CT2010 and the target flux is CT2011, both transported by FLEXPART. Temporally, there are systematic negative biases of the annual (blue squares in Fig. 4) and monthly (Fig. 5a) total flux estimates using the MCMC method, but they are relatively small compared to the difference between CT2010 and CT2011 (Table 2). For instance, the annual total posterior errors are about -8 and -3% for AB/SK and ON respectively using the most number of sub-regions. There is no linear relationship between the posterior error and the number of sub-regions using MCMC. The results using the seven spatial definitions for MCMC are marginally different. This characteristic of the MCMC inversion suggests that there is a limit to the number of sub-regions (or parameters) the MCMC method can optimize for a given inversion setup and constraining observations, and increasing this number of sub-regions may not improve the results. Synthetic data inversion like the present study is useful for evaluating the inversion setup to ensure that the (near) optimal number of parameters is used.

Unlike the MCMC method as shown in Fig. 4, posterior errors increase as the number of sub-regions increases in the western and eastern regions using the CFM method, similar to the previous case with only optimisation procedure error. Results suggest that the inherent optimisation error may have a dominant effect on the flux estimate even with only prior flux error. The annual and monthly biases change systematically from being negative to positive from 1 sub-region to 49 sub-regions (census divisions) in ON, similarly for AB/SK. It is interesting to note when the least number of sub-regions is used, the results of CFM and MCMC are quite comparable, but there appears to be seasonality in the monthly estimation errors using CFM as shown on the right column in Fig. 5a. This indicates that estimating many parameters in high-dimensional space is problematic for CFM. Divergent results appear when high-dimensional parameter space is involved in the inversion. Bielger et al. (2011) noted that parameter-estimation problem using minimization method in particular becomes extremely challenging even with relative few parameters to estimate. Although not shown, spatially, the annual patterns of the posterior fluxes obtained from the MCMC and the CFM methods are very similar to those of the targets. Unrealistic negative fluxes do not exist for the provinces of AB/SK and ON with any number sub-regions using either MCMC or CFM method.

The third (III) set of experiments E18–E24 (MCMC method) and E49–E55 (CFM method) represents conditions in which flux error does not exist, but there is transport model error which includes errors introduced by the simulations of the short term (5 days) transport and the baseline mole fractions (5 days previous) using the FLEXPART model. Similar to the previous set of experiments, it is important to keep in mind that the inherent optimisation procedure error exists. The target is the CT2011 model results at the 7 stations. Both FLEXPART and CarbonTracker models used CT2011 fossil fuel CO2 emissions as the prior fluxes.

The posterior errors have positive bias for the annual total flux estimates associated with both MCMC and CFM methods shown as green squares in Fig. 4a and b for AB/SK (western region) and ON (eastern region) respectively. The province of ON has relatively large error compared to the combined result of the two provinces in the western region. The annual flux bias does not have a linear relationship with the number of sub-regions (in contrast to the flux error case) using either inversion method. However, using the MCMC method with the largest number of sub-regions (E24), the annual flux estimates, are the least biased with 25 and 21 % for AB/SK and ON respectively. In addition, the associated standard deviations (YSTD) of the monthly biases for E24 are relatively small for the regions which means that the solution of the flux estimates is relatively stable. This appears to be a desirable result but the estimates of the individual sub-regions become unstable and some sub-regions have large errors.

In the CFM method, the annual estimation bias does not linearly increase as the number of sub-regions increases. This pattern appears to be the combination of the bias introduced by the transport error (similar to the MCMC results) and the increasing positive bias with the number of sub-regions from the optimisation procedure itself (shown in Sect. ). Comparing to the MCMC results, the biases of the annual total flux estimates using the largest number of sub-regions (E55) are 46 and 79 % for AB/SK and ON respectively. However, it is similar to the results using MCMC that the standard deviations (YSTD) of the monthly errors using the most number of sub-regions are relatively small except for AB/SK in which there is no significant difference in the annual errors using different number of sub-regions.

On the monthly time scale for the MCMC results as shown in Fig. 5b, there is a tendency of negative bias during the months of June–August (summer) for the provinces of AB/SK. Negative posterior errors only occur in warmer months of May–July for the province of ON using the MCMC method. The modelled time series of FLEXPART and CT2011 (Fig. 3) indicate that negative flux biases are due to the FLEXPART model over-predicting (compared to CT2011) the mole fractions at most of the stations being considered in the western region during the summer. Over-prediction in the mole fractions of fossil fuel CO2 subsequently leads to under-estimation (blue cells) in Tables S1–S4 in the posterior fluxes. The reversal behavior of over-estimation (red/orange/yellow cells) in the posterior fluxes occurs during the winter because of the under-prediction in the mole fractions in the cold season. As shown in Fig. 5b, the shape and the amplitude of seasonality of the monthly biases using the CFM method appear to be consistent with the MCMC method. The variability tends to reduce using the most number of sub-regions for both inversion methods.

Spatially, the annual patterns of the posterior fluxes obtained from the MCMC (not shown) and the CFM (not shown) methods appear to be similar to those of the targets. It is similar to set (II) of experiments in that, unrealistic negative fluxes appear in the province of ON using six or more sub-regions when either MCMC (not shown) or CFM (not shown) method is used, whereas negative fluxes do not appear for AB/SK regardless the number of sub-regions.

In summary, when only transport model error exists, the magnitude and variability of biases are relatively large independent of inversion methods compared to the previous set of experiments in which only flux error exists. This suggests that the accuracy of the posterior fluxes is more dependent on the modelled transport than on the prior fluxes in all the experiments we performed. Therefore the relative importance of this effect highlights the need of using the best possible transport model(s) for inversions. In absolute terms, the annual percentage errors are relatively small using the MCMC estimation method in comparison with the CFM estimation method.

The fourth (IV) set of experiments E25–E31 (MCMC method) and E56–E62 (CFM method) represents conditions in which both flux and transport model errors exist. Similar to the third (III) set of experiments, the posterior errors do not systematically decrease as the number of sub-regions increases in the MCMC method in the AB/SK and ON regions. As shown in Fig. 4a and b (black squares), the lack of dependence on the number of sub-regions and the similarity of the non-linear pattern compared to set (III) confirm the fact that the estimation bias are introduced and dominated by the transport model error. Transport model error often confounds inversion results and increases uncertainties. The combined (cancelling) effects of the prior flux and transport model errors are evident in the estimation biases as shown in the figures comparing to the previous two sets of experiments. Although the biases do not linearly depend on the number of sub-regions, in E31 which uses the most number of sub-regions, the biases are 4 and 16 % as shown in Fig. 4a and b for AB/SK and ON respectively which are the smallest in set E25–E31. To compare, the biases of E62 which has the same setup as E31 are 23 and 73 % for the AB/SK and ON regions when the CFM method is used. Figure 6a and b shows the linear regression analysis using all months of 2009 that plot modelled against synthetic observation fossil fuel CO2 using all thirty-seven and forty-nine sub-regions for AB/SK and ON respectively. The regression analyses of the prior and posterior CO2 mole fraction results are shown in blue and red respectively. The improvement of the fit in terms of R2 and the slope of the regression is the most substantial for the DOW station located in ON, which has the largest synoptic variability among all seven stations. Another indication that this station is the most affected by local fossil fuel CO2 sources. Note that stations EST in AB and DOW in ON have the lowest R2. All the inversion cases resulted in better slope and R2, but the posterior estimate biases as presented in Tables S1–S4 could be larger than the percentage difference of the prior and target fluxes (Table 2) which means the flux estimates are not necessarily better than the prior even with larger R2. Thus improvements in R2 in the posterior results are not necessarily a validation of the inversion results.

Figures 7 and 8 show spatially, the annual estimation errors (rightmost column) from the target flux for the seven spatial definitions over the provinces of AB/SK and ON respectively using the MCMC method. Blue and red areas represent under- and over-estimations compared to the target. There is a systematic underestimation (blue areas) using the MCMC method for the most part of AB/SK and ON, particularly for those areas that are far away from the stations and where the prior fluxes are relatively small. In other words, the variability (as a function of the number of sub-regions) of the spatial patterns appears to be limited to those areas where prior fluxes are large and close to the stations. Unrealistic negative posterior fluxes (black sub-regions) appear for the province of ON when six or more sub-regions (E28 to E31) are used which are consistent with previous sets of experiments. We note also that the posterior errors around the stations can be much larger than the percentage differences of the prior and the target fluxes, and there is no improvement with increasing sub-region resolution. This means that the accuracy of the inversion results is not simply determined by the sensitivity of the station's footprints, that high sensitivity does not guarantee reliable flux estimates. The posterior errors around stations are particularly noticeable using the CFM method that is to be discussed next.

Figures 9 and 10 show that the annual spatial patterns obtained from the CFM method are quite consistent with those obtained from the MCMC method, but the errors tend to be more positive possibly due to the inherent positive bias from the optimisation procedure as identified in Sect. . The posterior errors are not linearly dependent on the number of sub-regions using the CFM method. This might be caused by the non-linear nature of cost function in which the two terms in Eq. (4) are dependent on each other and the confounding impact of the transport error. However, the standard deviations (YSTD) of the monthly posterior errors are the least variable using the most number of sub-regions for both the western and eastern regions. This is consistent with the results using the MCMC method.

Results of a few sub-regions using the CFM estimation method show large monthly variability compared to the relatively small seasonality of the target fluxes (not shown). The spatial patterns of the monthly posterior error appear to be quite sporadic. Similar to the MCMC results, unrealistic negative posterior fluxes (black negative flux sub-regions) as shown in Figs. 9 and 10 appear for the province of ON when six or more sub-regions are used (E59 to E62). These unrealistic sub-regional fluxes are consistent between the two inversion methods and are similar to the transport error only case. It is important to point out again that the spatial patterns of the annual and monthly (not shown) posterior errors increase in similarity using the MCMC and CFM methods as the number of sub-regions decreases. The posterior flux estimates of the two inversion methods in fact become nearly identical when the least number of sub-regions are used. The spatial patterns (Figs. 9 and 10) indicate that the posterior errors do not change using the CFM method for sub-regions that are far away from the stations and in areas where the prior fluxes are relatively small, namely the northwestern AB and northern SK. There is a systematic underestimation (blue areas) using the CFM method for roughly the same sub-regions that are somewhat close to the stations of EGB and DOW in ON. However, different from the MCMC method, overestimations (red areas) occur in most part of AB/SK and ON that do not change noticeably from month to month (not shown).

In this set of experiments which can be considered to be similar to using real observations as constraint, on the annual time scale, there is a systematic positive bias in the posterior estimates using both MCMC and CFM methods. Note that the biases are relatively small using the MCMC method.

Lastly, we compare quantitatively the posterior flux estimates to the target on monthly and annual time scales. The monthly posterior fluxes and the probability distributions of the annual posterior fluxes in comparison with the targets are shown in Fig. 11 for the three provinces separately. The priors and targets are shown in gray and green for reference. This figure summarizes the results using experiments E28 and E27 as an example in which 11 and 4 sub-regions were used respectively for AB/SK and ON without producing unrealistic negative fluxes. These results are compared to experiment E31 in which all 37 and 49 sub-regions for AB/SK and ON were used respectively. Monthly flux estimates show large intra-annual variability compared to the target (green) fluxes for all three provinces. The annual means are significantly different (all t tests show p values less than 0.0001) between experiments E28 (E27) and E31 for all provinces. As shown in Fig.11a, noticeable error reductions occur in May for AB in which the 5th–95th percentile bands do not overlap. Similarly, there are noticeable error reductions in many months for ON as shown in Fig. 11c. In absolute terms, using the spatial definitions of E31, the means (red vertical lines) of the annual total flux estimates are 136, 36 and 161Mt for the provinces of AB, SK and ON respectively, which are significantly different from E28 and E27. These results which are relatively less biased in this set of experiments, compare well with the annual target (green vertical lines) fluxes of 130, 36 and 139Mt for AB, SK and ON respectively. The respective annual posterior errors of E31 for the three provinces are calculated as 5, 0 and 16 %, which compare well with the annual percentage differences of the CT2010 prior and CT2011 target fluxes for AB and SK with -26 and -24% respectively, but not for ON with 12 %. The worsening of the flux estimate results compared to the prior for ON is due to the poor modelled transport for some of the stations in this particular province as discussed in Sect. .

An important feature in Fig. 11 is that the monthly posterior uncertainty estimates (colored bands) could be incorrect (too small) as the uncertainties do not always cover the (sometimes large) differences between the posterior estimates and the target flux values. Another feature evident in Fig. 11 is the fluctuating nature of the inversion results over the limited variations examined in this study. It is clear that inversion results are strongly dependent on the inversion model setup, transport variations with time (different months and seasons) and spatial domain locations, etc. This could be a part of the reason for the widely different posterior flux estimates from different inversion studies with very different setups.

We will continue to investigate how the posterior uncertainty can be improved (more realistic) in our next set of synthetic data experiments examining the impact of different LPDM transport models, different background baseline mole-fraction estimation, observation site selections, etc.